17.2
Sonido como ondas de presión
Las ondas de sonido, que son ondas longitudinales, se pueden modelar como la amplitud de desplazamiento que varía en función de las coordenadas espaci…
Pensemos en el sonido viajando a través de un medio. Las perturbaciones longitudinales crean una diferencia de presión entre sus columnas sucesivas, que luego sufren oscilaciones.
Considere un cilindro no perturbado del medio, con un área de sección transversal A a lo largo del eje x. Su desplazamiento longitudinal, dado por y, es una función de onda.
A medida que la onda viaja, sus extremos en x-1 y x-2 son desplazados por y-1 e y-2, respectivamente. Si este último es mayor, el cilindro se expande y la presión cae de la presión circundante.
Se conoce su volumen inicial y se deriva su cambio de volumen. A continuación, se obtiene el cambio fraccional de volumen.
Recordemos la definición de módulo a granel, a partir del cual se obtiene la presión manométrica. Simplificando, se observa que la presión manométrica es una onda.
La presión manométrica es máxima en los puntos de desplazamiento cero y mínima en los puntos donde el desplazamiento es máximo.
Su amplitud es proporcional a la amplitud de desplazamiento, el módulo de volumen del medio y el número de onda. Por lo tanto, es inversamente proporcional a la longitud de onda.
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Q1: How do sound waves create pressure differences in a medium?
Sound waves are longitudinal disturbances that displace successive columns of a medium by different amounts. When one column displaces more than an adjacent column, the medium expands or compresses, creating a pressure difference from the surrounding pressure. This pressure variation propagates through the medium as the wave travels, forming the basis of sound wave behavior.
Q2: What is the relationship between displacement and gauge pressure in sound waves?
Gauge pressure in a sound wave is directly related to particle displacement through the medium's bulk modulus. Gauge pressure is maximum at points where particle displacement is zero (compression and rarefaction zones) and minimum where displacement is maximum. The pressure amplitude depends on displacement amplitude, bulk modulus, and wave number, making shorter wavelengths produce greater pressure amplitudes.
Q3: Why is gauge pressure zero at maximum displacement points?
At maximum displacement points, particles in the medium have moved farthest from their equilibrium positions but are not compressing or expanding relative to surrounding columns. This creates a neutral pressure state where gauge pressure equals zero. Compression occurs at zero displacement, producing maximum positive pressure, while rarefaction produces maximum negative pressure.
Q4: How does wavelength affect pressure amplitude in sound waves?
Pressure amplitude is inversely proportional to wavelength. Shorter wavelengths produce greater pressure amplitudes, while longer wavelengths produce smaller pressure amplitudes. This relationship arises because pressure amplitude depends on the wave number, which is inversely related to wavelength, making high-frequency sound waves generate larger pressure fluctuations.
Q5: What role does bulk modulus play in sound wave pressure?
Bulk modulus quantifies a medium's resistance to compression and directly determines gauge pressure from particle displacement. The relationship between instantaneous displacement and gauge pressure is derived through bulk modulus, which links the material's mechanical properties to pressure fluctuations. A higher bulk modulus produces greater pressure changes for the same displacement amplitude.
Q6: How do compression and rarefaction zones differ in pressure?
Compression zones occur where medium particles aggregate closely together, producing the most positive pressure. Rarefaction zones occur where particles are farthest apart, producing the most negative pressure. Between these zones, at maximum particle displacement, pressure returns to zero, creating the oscillating pressure pattern characteristic of sound waves.
Q7: What determines the amplitude of pressure fluctuations in sound waves?
Pressure amplitude is proportional to three factors: displacement amplitude, the bulk modulus of the medium, and the wave number. Since wave number is inversely proportional to wavelength, shorter wavelengths generate larger pressure amplitudes. These relationships show that stiffer materials and higher-frequency waves produce greater pressure variations.